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Beginning Partial Differential Equations, 2nd Edition

Beginning Partial Differential Equations, 2nd Edition

Peter V. O'Neil

ISBN: 978-1-118-03060-8

Oct 2011

496 pages

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A rigorous, yet accessible, introduction to partial differential equations—updated in a valuable new edition

Beginning Partial Differential Equations, Second Edition provides a comprehensive introduction to partial differential equations (PDEs) with a special focus on the significance of characteristics, solutions by Fourier series, integrals and transforms, properties and physical interpretations of solutions, and a transition to the modern function space approach to PDEs. With its breadth of coverage, this new edition continues to present a broad introduction to the field, while also addressing more specialized topics and applications.

Maintaining the hallmarks of the previous edition, the book begins with first-order linear and quasi-linear PDEs and the role of characteristics in the existence and uniqueness of solutions. Canonical forms are discussed for the linear second-order equation, along with the Cauchy problem, existence and uniqueness of solutions, and characteristics as carriers of discontinuities in solutions. Fourier series, integrals, and transforms are followed by their rigorous application to wave and diffusion equations as well as to Dirichlet and Neumann problems. In addition, solutions are viewed through physical interpretations of PDEs. The book concludes with a transition to more advanced topics, including the proof of an existence theorem for the Dirichlet problem and an introduction to distributions.

Additional features of the Second Edition include solutions by both general eigenfunction expansions and numerical methods. Explicit solutions of Burger's equation, the telegraph equation (with an asymptotic analysis of the solution), and Poisson's equation are provided. A historical sketch of the field of PDEs and an extensive section with solutions to selected problems are also included.

Beginning Partial Differential Equations, Second Edition is an excellent book for advanced undergraduate- and beginning graduate-level courses in mathematics, science, and engineering.

1. First Order Equations.

Notation and Terminology.

The Linear First Order Equation.

The Significance of Characteristics.

The Quasi-Linear Equation.

2. Linear Second Order Equations.


The Hyperbolic Canonical Form.

The Parabolic Canonical Form.

The Elliptic Canonical Form.

Some Equations of Mathematical Physics.

The Second Order Cauchy Problem.

Characteristics and the Cauchy Problem.

Characteristics As Carriers of Discontinuities.

3. Elements of Fourier Analysis.

Why Fourier Series?

The Fourier Series of a Function.

Convergence of Fourier Series.

Sine and Cosine Expansions.

The Fourier Integral.

The Fourier Transform.


Fourier Sine and Cosine Transforms.

4. The Wave Equation.

The Cauchy Problem and d'Alembert's Solution.

d'Alembert's Solution As a Sum of Waves.

The Characteristic Triangle.

The Wave Equation on a Half-Line.

A Problem on a Half-Line With Moving End.

A Nonhomogeneous Problem on the Real Line.

A General Problem on a Closed Interval.

Fourier Series Solutions on a Closed Interval.

A Nonhomogeneous Problem on a Closed Interval.

The Cauchy Problem by Fourier Integral.

A Wave Equation in Two Space Dimensions.

The Kirchhoff/Poisson Solution.

Hadamard's Method of Descent.

5. The Heat Equation.

The Cauchy Problem and Initial Conditions.

The Weak Maximum Principle.

Solutions on Bounded Intervals.

The Heat Equation on the Real Line.

The Heat Equation on the Half-Line.

The Debate Over the Age of the Earth.

The Nonhomogeneous Heat Equation.

The Heat Equation In Several Space Variables.

6. Dirichlet and Neumann Problems.

The Setting of the Problems.

Some Harmonic Functions.

Representation Theorems.

Two Properties of Harmonic Functions.

Is the Dirichlet Problem Well-Posed?

Dirichlet Problem for a Rectangle.

7. Existence Theorems.

A Classical Existence Theorem.

A Hilbert Space Approach.

Distributions and an Existence Theorem.

8. Additional Topics.

Solutions by Eigenfunction Expansions.

Numerical Approximations of Solutions.

Burger's Equation.

The Telegraph Equation.

Poisson's Equation.

9. End Materials.

Historical Notes.


Answers to Selected Exercises.

  • Sections on numerical approximation of solutions in Chapter 4 (The Wave Equation) and Chapter 5 (The Heat Equation) are new to the second edition.
  • A new chapter has been added that features selected applications and additional techniques, including solutions in series of orthogonal functions, convection/diffusion, shocks, shallow waves, as well as solitons and asymptotic solutions.
  • A second new chapter has been added and provides an introduction to the advanced theory of partial differential equations, including distributions, Hilbert spaces, Sobolev spaces, and weak solutions.
  • Solutions to selected problems are included at the end of the book
    and experimental, computer-based exercises are designed to develop students inquiries.  
  • A separate Student Solutions Manual is available
  •  Discussion of first order equations and the method of characteristics for quasi-linear first order PDEs; canonical forms of second order PDEs; characteristics and the Cauchy problem; a proof of the Cauchy-Kowalevski theorem for linear systems and connections between the mathematics and physical interpretations of PDEs.